Geometric Measure Theory and Real Analysis (Publications of the Scuola Normale Superiore Book 17)

In 2013, a school on Geometric Measure Theory and Real Analysis, organized by G. Alberti, C. De Lellis and myself, took place at the Centro De Giorgi in Pisa, with lectures by V. Bogachev, R. Monti, E. Spadaro and D. Vittone. The book collects the notes of the courses. The courses provide a deep and up to date insight on challenging mathematical problems and their recent developments: infinite-dimensional analysis, minimal surfaces and isoperimetric problems in the Heisenberg group, regularity of sub-Riemannian geodesics and the regularity theory of minimal currents in any dimension and codimension. 2.5 Excluding an infinite order of contact2.6 The persistence of singularities; 2.7 Sketch of the proof; 3 Q-valued functions and rectifiable currents; 3.1 Q-valued functions; 3.2 Graph of Lipschitz Q-valued functions; 3.3 Approximation of area minimizing currents; 4 Selection of contradiction's sequence; 5 Center manifold's construction; 5.1 Notation and assumptions; 5.2 Whitney decomposition and interpolating functions; 5.3 Normal approximation; 5.4 Construction criteria; 5.5 Splitting before tilting; 5.5.1 (HT)-cubes; 5.5.2 (Ex)-cubes; 5.6 Intervals of flattening; 5.6.1 Defining procedure 5.1 Existence and density estimates5.2 Examples of nonsmooth H-minimal surfaces; 5.2.1 A Lipschitz H-minimal surface; 5.2.2 An H-minimal intrinsic graph with discontinuous normal; 5.2.3 Sets with constant horizontal normal; 5.3 Lipschitz approximation and height estimate; References; Regularity of higher codimension area minimizing integral currents; 1 Introduction; 1.2 Partial regularity in higher codimension; 2 The blowup argument: a glimpse of the proof; 2.1 Flat tangent cones do not imply regularity; 2.2 Non-homogeneous blowup; 2.3 Multiple valued functions; 2.4 The need of centering Isoperimetric problem and minimal surfaces in the Heisenberg group1 Introduction to the Heisenberg group Hn; 1.1 Algebraic structure; 1.2 Metric structure; 2 Heisenberg perimeter and other equivalent measures; 2.1 H-perimeter; 2.2 Equivalent notions for H-perimeter; 2.2.1 Hausdorff measures; 2.2.2 Minkowski content and H-perimeter; 2.2.3 Integral differential quotients; 2.3 Rectifiability of the reduced boundary; 3 Area formulas, first variation and H-minimal surfaces; 3.1 Area formulas; 3.1.1 Sets with Lipschitz boundary; 3.1.2 Formulas for the horizontal inner normal 3.1.3 Area formula for t-graphs3.1.4 Area formula for intrinsic graphs; 3.2 First variation and H-minimal surfaces; 3.2.1 First variation of the area for t-graphs; 3.2.2 Characteristic points; 3.2.3 First variation of the area functional for intrinsic graphs; 3.3 First variation along a contact flow; 4 Isoperimetric problem; 4.1 Existence of isoperimetric sets and Pansu's conjecture; 4.2 Isoperimetric sets of class C2; 4.3 Convex isoperimetric sets; 4.4 Axially symmetric solutions; 4.5 Calibration argument; 5 Regularity problem for H-perimeter minimizing sets Cover; Title Page; Copyright Page; Table of Contents; Preface; Sobolev classes on infinite-dimensional spaces; Introduction; 1 Measures on infinite-dimensional spaces; 2 Gaussian measures; 3 Integration by parts and differentiable measures; 4 Sobolev classes over Gaussian measures; 5 Inequalities and embeddings; 6 Sobolev classes over differentiable measures; 7 The class BV: the Gaussian case; 8 The class BV: the general case; 9 Sobolev functions on domains and their extensions; 10 BV functions on domains and their extensions; References Sobolev Classes On Infinite-dimensional Spaces / Vladimir I. Bogachev -- Isoperimetric Problem And Minimal Surfaces In The Heisenberg Group / Roberto Monti -- Regularity Of Higher Codimension Area Minimizing Integral Currents / Emanulele Spadaro -- The Regularity Problem For Sub-riemannian Geodesics / Davide Vittone. Edited By Luigi Ambrosio. Includes Bibliographical References. Front Matter....Pages i-ix Sobolev classes on infinite-dimensional spaces....Pages 1-56 Isoperimetric problem and minimal surfaces in the Heisenberg group....Pages 57-129 Regularity of higher codimension area minimizing integral currents....Pages 131-192 The regularity problem for sub-Riemannian geodesics....Pages 193-226 Back Matter....Pages 227-228